Bug on band

An infinitely stretchable rubber band has one end nailed to a wall, while the other end is pulled away from the wall at the rate of 1 m/s; initially the band is 1 meter long. A bug on the rubber band, initially near the wall end, is crawling toward the other end at the rate of 0.001 cm/s. Will the bug ever reach the other end? If so, when?

2. Relevant equations
Differential ones!

3. The attempt at a solution
I solved this one in what I think is a sort of novel way. I imagine that we are viewing the situation in a "stretching" frame--as if we view the stretching band with a camera and we continuously zoom out to keep the image of the band on the film exactly 1 meter wide. Then, the velocity of the bug on the film is described by

[tex]v_{bug\,in\,frame} = \frac{l_0}{l} v_{bug\,0}[/tex]

where [itex]l_0[/itex] is the initial length of the band, [itex]l[/itex] is the length of the band as a function of time, and [itex]v_{bug\,0}[/itex] is the initial velocity of the bug as seen on the film (which is the same as the real velocity, 1E-5 m/s).

In effect, the 'image on the film' becomes a representation for the fraction of the band traversed by the bug.

So, using the information given in the problem, this equation becomes

[tex]v_{bug\,in\,frame} = \frac{1}{1+t} 1 \times 10^{-5}[/tex]

which we can integrate with respect to time to find the x position of the bug on the film:

[tex]\int v\,dt = \int \frac{1}{1+t} 1 \times 10^{-5}\,dt[/tex]

[tex]x = 1 \times 10^{-5} \ln(1+t)[/tex]

When x = 1, the bug has reached the end of the band:

[tex]1 = 1 \times 10^{-5} ln(1+t)[/tex], and with a little algebra,

[tex]t = e^{100000} - 1[/tex].

My original plan, however, was to use the following differential equation to describe the actual distance of the bug from the wall:

[tex]\frac{dx}{dt} = \frac{x}{l} v_{end} + v_{bug}[/tex]

where [itex]l[/itex] (which is 1+t) is a function of t that describes the length of the band, [itex]v_{end}[/itex] is the velocity of the end of the band (dl/dt=1 m/s) and [itex]v_{bug}[/itex] is the velocity of the bug by itself (1E-5 m/s). But I didn't know how to solve this differential equation (I haven't taken a diff eq course yet and separation of variables won't work). I'm wondering--is there a general solution to this form of DE?